In many practical settings, participants are willing to deviate
from the protocol only if they remain undetected. Aumann and Lindell
introduced a concept of covert adversaries to formalize this type of
corruption. In the current paper, we refine their model to get stronger
security guarantees. Namely, we show how to construct protocols, where
malicious participants cannot learn anything beyond their intended output
and honest participants can detect malicious behavior that alters their
outputs. As this construction does protect honest parties from selective
protocol failures, a valid corruption complaint can leak a single bit of
information about the inputs of honest parties. Importantly, it is often up
to the honest party to decide whether to complain or not. Moreover, in
practice, this potential leakage is often outweighed by the efficiency
compared to standard zero-knowledge based proofs. As a concrete practical
contribution, we show how to implement consistent versions of several
important cryptographic protocols such as oblivious transfer, conditional
disclosure of secrets and private inference control..